Advertisements
Advertisements
Question
Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0
Advertisements
Solution
sec2y tan x dy + sec2x tan y dx = 0
Dividing both sides by tan x tan y, we get
`(sec^2y tan x)/(tanx tan y) "d"y + (sec^2x tany)/(tanx tany) "d"x` = 0
∴ `(sec^2x)/(tanx) "d"x + (sec^2y)/(tany) "d"y` = 0
Integrating on both sides, we get
`int (sec^2x)/(tanx) "d"x + int (sec^2y)/(tany) "d"y` = 0
∴ log |tan x| + log |tan y| = log |c|
∴ log |tan x.tan y| = log |c|
∴ tan x tan y = c
APPEARS IN
RELATED QUESTIONS
If 1, `omega` and `omega^2` are the cube roots of unity, prove `(a + b omega + c omega^2)/(c + s omega + b omega^2) = omega^2`
Verify that y2 = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]
Show that Ax2 + By2 = 1 is a solution of the differential equation x \[\left\{ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right\} = y\frac{dy}{dx}\]
Verify that \[y = ce^{tan^{- 1}} x\] is a solution of the differential equation \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + \left( 2x - 1 \right)\frac{dy}{dx} = 0\]
Show that y = e−x + ax + b is solution of the differential equation\[e^x \frac{d^2 y}{d x^2} = 1\]
xy (y + 1) dy = (x2 + 1) dx
x cos2 y dx = y cos2 x dy
tan y dx + sec2 y tan x dy = 0
Solve the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + \left( 1 + y^2 \right) = 0\], given that y = 1, when x = 0.
Find the particular solution of edy/dx = x + 1, given that y = 3, when x = 0.
y ex/y dx = (xex/y + y) dy
Solve the following differential equations:
\[\frac{dy}{dx} = \frac{y}{x}\left\{ \log y - \log x + 1 \right\}\]
Solve the following initial value problem:-
\[dy = \cos x\left( 2 - y\text{ cosec }x \right)dx\]
The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 years, and the present population is 100000, when will the city have a population of 500000?
Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?
Find the equation of the curve passing through the point (0, 1) if the slope of the tangent to the curve at each of its point is equal to the sum of the abscissa and the product of the abscissa and the ordinate of the point.
The solution of the differential equation y1 y3 = y22 is
The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is
The differential equation satisfied by ax2 + by2 = 1 is
Verify that the function y = e−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
`y=sqrt(a^2-x^2)` `x+y(dy/dx)=0`
If a + ib = `("x" + "iy")/("x" - "iy"),` prove that `"a"^2 +"b"^2 = 1` and `"b"/"a" = (2"xy")/("x"^2 - "y"^2)`
Choose the correct option from the given alternatives:
The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is
For the following differential equation find the particular solution.
`dy/ dx = (4x + y + 1),
when y = 1, x = 0
Solve the following differential equation.
`dy/dx + y = e ^-x`
Solve the following differential equation.
`dy/dx + 2xy = x`
Solve the following differential equation.
dr + (2r)dθ= 8dθ
Select and write the correct alternative from the given option for the question
The differential equation of y = Ae5x + Be–5x is
Solve `("d"y)/("d"x) = (x + y + 1)/(x + y - 1)` when x = `2/3`, y = `1/3`
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
Solve the differential equation `"dy"/"dx"` = 1 + x + y2 + xy2, when y = 0, x = 0.
Solve the differential equation `dy/dx + xy = xy^2` and find the particular solution when y = 4, x = 1.
